Multiple integral problem (limits interchanged)

Question

I would appreciate the help if someone explains how can we get the result of the integral $$ \int_0^L m\ddot{u}(x)\left[\int_0^x \frac{\partial^2 w(\xi)}{\partial \xi^2} d\xi\right] dx = \int_0^L \left[\int_{\xi}^L m\ddot{u}(x)dx \right]\frac{\partial^2w}{\partial\xi^2}d\xi $$

Answer 1

Let's make a quick sketch of the domain of the integration. The horizontal axis is $x$, the vertical axis is $\xi$. In the first integral, $x$ goes from $0$ to $L$ (8 in the picture). The integration along $\xi$ is the red line, that goes from $0$ to $x$. In the second case, where we reverse the order of integration, $\xi$ is between $0$ and $L$ (8 in the vertical direction), and then $x$ is integrated along the green line, from $\xi$ to $L$,

Answer 2

It comes from the traditional variable switch trick from multivariate calculus

The region $$0 \leq x \leq L, 0 \leq \xi \leq x$$ is the same as the region $$0 \leq \xi \leq L, \xi \leq x \leq L$$